This question is a follow-up of an old question posted on MathOverflow.

**Motivation:** The exact equations of modular curves $X_0^{+}(p)=X_0(p)/w_p$($p>13$ is a prime number,$w_p$ is the Fricke involution) are intriguing objects. One can construct exact equations from modular forms of weight 2 on $\Gamma_0(p)$ from a quaternion algebra $A(p)$ over $\mathbb{Q}$ ramified at $p$ and $\infty$. Let $O$ be a maximal order of $A(p)$ and $I$ be a left $O$-ideal. The theta function determined by $I$ $$\theta_I(\tau)=\sum_{x\in I}e^{2\pi i\tau\frac{N(x)}{N(I)}}$$ is always a modular form of weight 2 on $\Gamma_0(p)$ with Fricke eigenvalue $-1$(see A. Pizer's paper).

**Experiment:** According to the Proposition 2.17 of Pizer's paper, the number of different theta functions determined by left ideals of $A(p)$ is bounded by the *type number* $T(p)$, which can be computed explicitly in another paper of Pizer (p. 94). A numerical computation with MAGMA seems to suggest that the number of distinct theta functions associated to the left $O$-ideals of $A(p)$ always reaches the upper bound, and all these theta functions are linear independent.

**Question:** Is there any example that the linear independence of theta functions fails for a certain prime number $p>13$? If not, is there a reference for the proof of this fact?

**Update:** The conjecture is likely to be false for $p=227$.